The Distributive Property, Part 2

5 min

Narrative

Students reason about the area of a rectangle with a variable side length. They express the area for different values of the variable and when the value is unknown. Students review symbolic notation for showing multiplication as they express the product of a number and a variable. The reasoning here will be helpful later in the lesson when students apply the distributive property in the context of finding the areas of rectangles whose side lengths are expressions with variables.

Launch

Allow students 2–3 minutes of quiet work time, followed by a whole-class discussion.

Student Task

  1. A rectangle has a length of 4 units and a width of mm units. Write an expression for the area of this rectangle.

  2. What is the area of the rectangle if mm is:

    3 units?

    2.2 units?

    15\frac15 unit?

  3. Could the area of this rectangle be 11 square units? Explain your reasoning.

Sample Response

  1. 4m4m (or equivalent)
  2. 12 square units, 8.8 square units, 45\frac45 square unit
  3. Yes, the area could be 11 square units. Sample reasoning: mm would have to be 114\frac{11}{4} units, since 4⋅114=114 \boldcdot \frac{11}{4}=11.

Synthesis

Select students to share their response to each question. Consider displaying a diagram of a rectangle and annotating it to illustrate students’ responses or explanations. Highlight the following points:

  • Rectangle areas can be found by multiplying length by width.
  • Both 4m4m and m⋅4m \boldcdot 4 are expressions for the area of this rectangle. These are equivalent expressions.
  • Lengths don’t have to be whole numbers. Neither do areas.
Anticipated Misconceptions

If students indicate they are not sure how to start and haven't drawn a diagram of a rectangle, suggest that they do so.

Standards
Addressing
  • 6.EE.2·Write, read, and evaluate expressions in which letters stand for numbers.
  • 6.EE.A.2·Write, read, and evaluate expressions in which letters stand for numbers.

10 min

20 min